Hamiltonian

canonical momentum

\[p = \frac{\partial \mathcal{L}}{\partial \dot{q}}\] \[H(p,q) = \sum_i p_i\dot{q}_i - L(q,\dot{q})\] \[\frac{\text{d}q^i}{\text{d}t}=\frac{\partial H}{\partial p_i}\] \[\frac{\text{d}p_i}{\text{d}t}=-\frac{\partial H}{\partial p^i}\]